Spatial processes and statistical modelling

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Spatial processes and statistical modelling

• Peter Green
• University of Bristol, UK
• BCCS GMCSS 2008/09 Lecture 8

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Spatial indexing

• Continuous space
• Discrete space
• lattice
• irregular - general graphs
• areally aggregated
• Point processes
• other object processes

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Space vs. time

• apparently slight difference
• profound implications for mathematical
  formulation and computational tractability

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Requirements of particular application domains

• agriculture (design)
• ecology (sparse point pattern, poor data?)
• environmetrics (space/time)
• climatology (huge physical models)
• epidemiology (multiple indexing)
• image analysis (huge size)

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Key themes

• conditional independence
• graphical/hierarchical modelling
• aggregation
• analysing dependence between differently indexed
  data
• opportunities and obstacles
• literal credibility of models
• Bayes/non-Bayes distinction blurred

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Why build spatial dependence into a model?

• No more reason to suppose independence in
  spatially-indexed data than in a time-series
• However, substantive basis for form of spatial
  dependent sometimes slight - very often space is
  a surrogate for missing covariates that are
  correlated with location

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Discretely indexed data
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Modelling spatial dependence in
discretely-indexed fields

• Direct
• Indirect
• Hidden Markov models
• Hierarchical models

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Hierarchical models, using DAGs

• Variables at several levels - allows modelling of
  complex systems, borrowing strength, etc.

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Modelling with undirected graphs

• Directed acyclic graphs are a natural
  representation of the way we usually specify a
  statistical model - directionally
• disease ? symptom
• past ? future
• parameters ? data
• whether or not causality is understood.
• But sometimes (e.g. spatial models) there is no
  natural direction

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Conditional independence

• In model specification, spatial context often
  rules out directional dependence (that would have
  been acceptable in time series context)

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Conditional independence

• In model specification, spatial context often
  rules out directional dependence

X20
X21
X22
X23
X24
X10
X11
X12
X13
X14
X00
X01
X02
X03
X04
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Conditional independence

• In model specification, spatial context often
  rules out directional dependence

X20
X21
X22
X23
X24
X10
X11
X12
X13
X14
X00
X01
X02
X03
X04
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Directed acyclic graph
a
b
c
in general
d
for example
p(a,b,c,d)p(a)p(b)p(ca,b)p(dc)
In the RHS, any distributions are legal, and
uniquely define joint distribution
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Undirected (CI) graph
Regular lattice, irregular graph, areal data...
X20
X21
X22
Absence of edge denotes conditional independence
given all other variables
X10
X11
X12
X00
X01
X02
But now there are non-trivial constraints on
conditional distributions
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Undirected (CI) graph
Suppose we assume
(?)
X20
X21
X22
X10
X11
X12
then
clique
X00
X01
X02
and so
The Hammersley-Clifford theorem says essentially
that the converse is also true - the only sure
way to get a valid joint distribution is to use
(?)
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Hammersley-Clifford
A positive distribution p(X) is a Markov random
field
X20
X21
X22
X10
X11
X12
if and only if it is a Gibbs distribution
X00
X01
X02
- Sum over cliques C (complete subgraphs)
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Partition function
Almost always, the constant of proportionality in
X20
X21
X22
X10
X11
X12
is not available in tractable form an obstacle
to likelihood or Bayesian inference about
parameters in the potential functions Physicists
call the partition function
X00
X01
X02
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Markov properties for undirected graphs

• The situation is a bit more complicated than it
  is for DAGs. There are 4 kinds of Markovness
• P pairwise
• Non-adjacent pairs of variables are conditionally
  independent given the rest
• L local
• Conditional only on adjacent variables
  (neighbours), each variable is independent of all
  others

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• G global
• Any two subsets of variables separated by a third
  are conditionally independent given the values of
  the third subset.
• F factorisation
• the joint distribution factorises as a product of
  functions of cliques
• In general these are different, but F?G?L?P
  always. For a positive distribution, they are all
  the same.

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Gaussian Markov random fields spatial
autoregression
If VC(XC) is -?ij(xi-xj)2/2 for Ci,j and 0
otherwise, then
is a multivariate Gaussian distribution, and
is the univariate Gaussian distribution
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A B C D
Gaussian random fields
A B C D
Inverse of (co)variance matrix dependent case
A
B
C
D
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Non-Gaussian Markov random fields
Pairwise interaction random fields with less
smooth realisations obtained by replacing squared
differences by a term with smaller tails, e.g.
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Discrete-valued Markov random fields
Besag (1974) introduced various cases of
for discrete variables, e.g. auto-logistic
(binary variables), auto-Poisson (local
conditionals are Poisson), auto-binomial, etc.
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Auto-logistic model
(Xi 0 or 1)
- a very useful model for dependent binary
variables (NB various parameterisations)
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Statistical mechanics models
The classic Ising model (for ferromagnetism) is
the symmetric autologistic model on a square
lattice in 2-D or 3-D. The Potts model is the
generalisation to more than 2 colours
and of course you can usefully un-symmetrise this.
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Auto-Poisson model
For integrability, ?ij must be ?0, so this
only models negative dependence very limited use.
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Hierarchical models and hidden Markov processes
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Chain graphs

• If both directed and undirected edges, but no
  directed loops
• can rearrange to form global DAG with undirected
  edges within blocks

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Chain graphs

• If both directed and undirected edges, but no
  directed loops
• can rearrange to form global DAG with undirected
  edges within blocks
• Hammersley-Clifford within blocks

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Hidden Markov random fields

• We have a lot of freedom modelling
  spatially-dependent continuously-distributed
  random fields on regular or irregular graphs
• But very little freedom with discretely
  distributed variables
• ? use hidden random fields, continuous or
  discrete
• compatible with introducing covariates, etc.

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Hidden Markov models
e.g. Hidden Markov chain
z0
z1
z2
z3
z4
hidden
y1
y2
y3
y4
observed
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Hidden Markov random fields
Unobserved dependent field
Observed conditionally-independent discrete field
(a chain graph)
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Spatial epidemiology applications
relative risk
expected cases
cases

• independently, for each region i. Options
• CAR, CARwhite noise (BYM, 1989)
• Direct modelling of ,e.g. SAR
• Mixture/allocation/partition models
• Covariates, e.g.

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Spatial epidemiology applications
Spatial contiguity is usually somewhat idealised
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Spatial epidemiology applications

• Richardson Green (JASA, 2002) used a hidden
  Markov random field model for disease mapping

observed incidence
relative risk parameters
expected incidence
hidden MRF
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Chain graph for disease mapping
based on Potts model
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Larynx cancer in females in France
SMRs